This Week's Finds in Mathematical Physics (Week 239)

August 16, 2006
This Week's Finds in Mathematical Physics (Week 239)
John Baez

David Corfield, Urs Schreiber and I have started up a new blog!

David is a philosopher, Urs is a physicist, and I'm a mathematician,
but one thing we all share is a fondness for n-categories. We
also like to sit around and talk shop in a public place where our
friends can drop by. Hence the title of our blog:

Technologically speaking, the cool thing about this blog is that it
uses itex and MathML to let us (and you) write pretty equations in TeX.
For this we thank Jacques Distler, who pioneered the technology on
his own blog:

Urs began by posting about 11d supergravity and higher gauge theory
(see "week237"). Now he's discussing Barrett and Connes' new work
on the Standard Model. Meanwhile, I've been obsessed with the
categorical semantics of quantum computation, and David has been
running discussions on categorifying Klein's Erlangen program (see
"week213"), the differences between mathematicians and historians
when it comes to writing histories of math, and so on.

And, it's all free.

Meanwhile, in the bad old world of extortionist math publishers,
we see a gleam of hope. The entire editorial board of the journal
Topology resigned to protest Reed-Elsevier's high prices!

The board includes some topologists I respect immensely. It takes
some guts for full-fledged memmbers of the math establishment to
do something like this, and I congratulate them for it. It'll be
fun to see what stooges Reed-Elsevier rounds up to form a new board
of editors. I can't imagine they'll just declare defeat and let the
journal fold.

This is part of trend where journal editors "declare independence"
from their publishers and move toward open access:

These notes are from an exciting period in physics, shortly after
the 1947 Shelter Island conference where Feynman and Schwinger
presented their approaches to quantum electrodynamics to an audience
of luminaries including Bohr, Oppenheimer, von Neumann, and Weisskopf.
Nobody understood Feynman's diagrams except Schwinger and maybe
Feynman's thesis advisor, John Wheeler.

Every true fan of physics loves reading about this heroic era and
its figures, especially Feynman. So, if you haven't read these yet,
run to the bookstore and buy them now!

6) James Gleick, Genius: the Life and Science of Richard Feynman,
Vintage Press, 1993.

7) Jagdish Mehra, The Beat of a Different Drum: the Life and Science
of Richard Feynman, Oxford U. Press, 1996.

8) Silvan S. Schweber, QED and the Men Who Made It, Princeton U.
Press, Princeton, 1994.

The first book is a barrel of fun but doesn't get into the nitty-gritty
details of Feynman's work. The second more scholarly treatment also
has lots of Feynman anecdotes - even some new ones! But, it covers
his work in enough detail to intimidate any non-physicist. The third
offers a broader panorama of the development of quantum electrodynamics.
Taken together, they add up to quite a nice story.

10) Richard P. Feynman, What Do *You* Care What Other People Think?
(Further Adventures of a Curious Character), W. W. Norton and Company,
New York, 2001.

They're more fun than everything else I've ever recommended on This
Week's Finds, combined. If you haven't read them, don't just *run* to
the nearest bookstore - get in a time machine, go back, and make sure
you *did* read them.

Today I'd like to wrap up the discussion of Koszul duality which I
began last Week. As we'll see, this gives a really efficient way
of categorifying the theory of Lie algebras and defining "Lie
n-algebras". And, as Urs Schreiber notes, these seem to be just
what we need to understand 11-dimensional supergravity in a nice
geometric way.

But before I dive into this heavy stuff, something fun. Thanks to
Christine Dantas' blog, I just saw a webpage on the origins of math
and writing in Mesopotamia:

These are little geometric clay figures that represented things like
sheep, jars of oil, and various amounts of grain. They are found
throughout the Near East starting with the agricultural revolution in
about 8000 BC. Apparently they were used for contracts! Eventually
groups of them were sealed in clay envelopes, so any attempt to tamper
with them would be visible.

But, it's annoying to have to break a clay envelope just to see what's
in it. So, after a while, they started marking the envelopes to say
what was inside.

Later, these marks were simply drawn on tablets. Eventually they gave
up on the tokens - a triumph of convenience over security. The marks
on tablets then developed into the Babylonian number system! The
transformation was complete by 3000 BC.

So, five millennia of gradual abstraction led to the writing of numbers!
From three tokens representing jars of oil, we eventually reach the
abstract number "3" applicable to anything.

Of course, all history is detective work. The story I just told is
an interpretation of archaeological evidence. It could be wrong.
This particular interpretation is due to Denise Schmandt-Besserat.
It seems to be fairly well accepted in broad outline, but scholars
are still arguing about it.

For more on her ideas, try this:

13) Denise Schmandt-Besserat, Accounting with tokens in the
ancient Near East,
http://www.utexas.edu/cola/centers/lrc/numerals/dsb/dsb.html [Broken]

For a bibliography of her many papers, try:

14) Denise Schmandt-Besserat, Publications,
http://www.utexas.edu/cola/centers/lrc/iedocctr/ie-pubs/dsb-pubs.html [Broken]
For more work on this subject - I want to read more! - try:

From the distant past, let's now shoot straight into the 20th
century. Last week I gave three examples of Koszul duality:

Making the free graded-commutative algebra on SL* into a differential
graded-commutative algebra is the same as making L into a Lie algebra.

Making the free graded Lie algebra on SL* into a differential
graded Lie algebra is the same as making L into a commutative algebra.

Making the free graded associative algebra on SL* into a differential
graded associative algebra is the same as making L into an associative
algebra.

Here L is a vector space, which we think of as a graded vector space
concentrated in degree zero. L* is its dual, and SL* is the "shifted"
or "suspended" version of L*, where we add one to the degree of
everything.

Now, what if we replace L by a graded vector space that can have stuff
of any degree? We get a fancier version of Koszul duality, which goes
like this:

Making the free graded-commutative algebra on SL* into a differential
graded-commutative algebra is the same as making L into an L-infinity
algebra.

Making the free graded Lie algebra on SL* into a differential
graded Lie algebra is the same as making L into a C-infinity algebra.

Making the free graded associative algebra on SL* into a differential
graded associative algebra is the same as making L into an A-infinity
algebra.

Here an "L-infinity algebra" is a chain complex that's like a Lie
algebra, except the Jacobi identity holds up to a chain homotopy called
the "Jacobiator", which in turn satisfies its own identity up to a
chain homotopy called the "Jacobiatorator", and so on ad infinitum.
Keeping track of all these higher homotopies is quite a chore. Well,
it's sort of fun when you get into it, but the great thing about
Koszul duality is that you don't need to remember any fancy formulas:
all the higher homotopies are packed into the *differential* on SL*.

Similarly, a "C-infinity algebra" is a chain complex that's like a
graded-commutative algebra up to homotopy, ad infinitum.

Similarly, an "A-infinity algebra" is a chain complex that's like an
associative algebra up to homotopy, ad infinitum. Here you can read off
all the higher homotopies from the Stasheff associahedra, which you
know and love from "week144" - but again, Koszul duality means you
don't have to!

As mentioned last week, all this stuff generalizes to any kind of
algebraic gadget in Vect - the category of vector spaces - which is
defined by a "quadratic operad" O. Any such operad has a "Koszul
dual" operad O* such that:

Making the free graded O-algebra on SL* into a differential
graded O-algebra is the same as making L into an O-infinity algebra.

Here O-infinity is an operad in the category of chain complexes
defined by "weakening" O in a systematic way - replacing all the
laws by chain homotopies, ad infinitum. We can define O-infinity
using the "bar construction", as nicely described here:

See "week191" for more on this book, and what the heck an "operad"
is.

Anyway, I don't have much intuition for how Koszul duality lets
us magically sidestep the bar construction of O-infinity - someday
I hope I'll understand this.

But, once we have the concept of "L-infinity algebra", we can
restrict ourselves to chain complexes that vanish except for their
first n terms - that is, degrees 0, 1, ..., n-1 - and get the
concept of "Lie n-algebra".

In fact, a Lie n-algebra is like a hybrid of a Lie algebra and an
n-category! The definition I just gave says a Lie n-algebra is
an L-infinity algebra which as a chain complex vanishes above
degree n-1. But, such chain complexes are equivalent to strict
n-category objects in Vect! So, we can think of Lie n-algebras as
strict n-categories that do their best to act like Lie algebras, but
with all the laws holding up to isomorphism, with these isomorphisms
satisfying their own laws up to isomorphism, etcetera.

But, the really cool part is that we can do *gauge theory* using
Lie n-algebras instead of Lie algebra, and taking n = 3 we get an
example that seems to explain the geometry of 11d supergravity...
that is, the classical limit of that mysterious thing called M-theory.

I never once doubted that I would eventually succeed in getting to the
bottom of things. - Alexander Grothendieck

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